The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__p(s(0)) → 0
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Rewrite Strategy: INNERMOST

(1) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to weighted TRS

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

a__f(0) → cons(0, f(s(0))) [1]
a__f(s(0)) → a__f(a__p(s(0))) [1]
a__p(s(0)) → 0 [1]
mark(f(X)) → a__f(mark(X)) [1]
mark(p(X)) → a__p(mark(X)) [1]
mark(0) → 0 [1]
mark(cons(X1, X2)) → cons(mark(X1), X2) [1]
mark(s(X)) → s(mark(X)) [1]
a__f(X) → f(X) [1]
a__p(X) → p(X) [1]

Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

a__f(0) → cons(0, f(s(0))) [1]
a__f(s(0)) → a__f(a__p(s(0))) [1]
a__p(s(0)) → 0 [1]
mark(f(X)) → a__f(mark(X)) [1]
mark(p(X)) → a__p(mark(X)) [1]
mark(0) → 0 [1]
mark(cons(X1, X2)) → cons(mark(X1), X2) [1]
mark(s(X)) → s(mark(X)) [1]
a__f(X) → f(X) [1]
a__p(X) → p(X) [1]

The TRS has the following type information:

a__f :: 0:s:f:cons:p → 0:s:f:cons:p 0 :: 0:s:f:cons:p cons :: 0:s:f:cons:p → 0:s:f:cons:p → 0:s:f:cons:p f :: 0:s:f:cons:p → 0:s:f:cons:p s :: 0:s:f:cons:p → 0:s:f:cons:p a__p :: 0:s:f:cons:p → 0:s:f:cons:p mark :: 0:s:f:cons:p → 0:s:f:cons:p p :: 0:s:f:cons:p → 0:s:f:cons:p

Rewrite Strategy: INNERMOST

(5) CompletionProof (UPPER BOUND(ID) transformation)

The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

a__p
mark
a__f

Due to the following rules being added:
none

And the following fresh constants: none

(6) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a__f(0) → cons(0, f(s(0))) [1] a__f(s(0)) → a__f(a__p(s(0))) [1] a__p(s(0)) → 0 [1] mark(f(X)) → a__f(mark(X)) [1] mark(p(X)) → a__p(mark(X)) [1] mark(0) → 0 [1] mark(cons(X1, X2)) → cons(mark(X1), X2) [1] mark(s(X)) → s(mark(X)) [1] a__f(X) → f(X) [1] a__p(X) → p(X) [1] The TRS has the following type information: a__f :: 0:s:f:cons:p → 0:s:f:cons:p 0 :: 0:s:f:cons:p cons :: 0:s:f:cons:p → 0:s:f:cons:p → 0:s:f:cons:p f :: 0:s:f:cons:p → 0:s:f:cons:p s :: 0:s:f:cons:p → 0:s:f:cons:p a__p :: 0:s:f:cons:p → 0:s:f:cons:p mark :: 0:s:f:cons:p → 0:s:f:cons:p p :: 0:s:f:cons:p → 0:s:f:cons:p Rewrite Strategy: INNERMOST

(7) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(8) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a__f(0) → cons(0, f(s(0))) [1] a__f(s(0)) → a__f(0) [2] a__f(s(0)) → a__f(p(s(0))) [2] a__p(s(0)) → 0 [1] mark(f(f(X'))) → a__f(a__f(mark(X'))) [2] mark(f(p(X''))) → a__f(a__p(mark(X''))) [2] mark(f(0)) → a__f(0) [2] mark(f(cons(X1', X2'))) → a__f(cons(mark(X1'), X2')) [2] mark(f(s(X3))) → a__f(s(mark(X3))) [2] mark(p(f(X4))) → a__p(a__f(mark(X4))) [2] mark(p(p(X5))) → a__p(a__p(mark(X5))) [2] mark(p(0)) → a__p(0) [2] mark(p(cons(X1'', X2''))) → a__p(cons(mark(X1''), X2'')) [2] mark(p(s(X6))) → a__p(s(mark(X6))) [2] mark(0) → 0 [1] mark(cons(X1, X2)) → cons(mark(X1), X2) [1] mark(s(X)) → s(mark(X)) [1] a__f(X) → f(X) [1] a__p(X) → p(X) [1] The TRS has the following type information: a__f :: 0:s:f:cons:p → 0:s:f:cons:p 0 :: 0:s:f:cons:p cons :: 0:s:f:cons:p → 0:s:f:cons:p → 0:s:f:cons:p f :: 0:s:f:cons:p → 0:s:f:cons:p s :: 0:s:f:cons:p → 0:s:f:cons:p a__p :: 0:s:f:cons:p → 0:s:f:cons:p mark :: 0:s:f:cons:p → 0:s:f:cons:p p :: 0:s:f:cons:p → 0:s:f:cons:p Rewrite Strategy: INNERMOST

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

0 => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
a__f(z) -{ 2 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
a__f(z) -{ 1 }→ 1 + 0 + (1 + (1 + 0)) :|: z = 0
a__p(z) -{ 1 }→ 0 :|: z = 1 + 0
a__p(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
mark(z) -{ 2 }→ a__p(a__p(mark(X5))) :|: X5 >= 0, z = 1 + (1 + X5)
mark(z) -{ 2 }→ a__p(a__f(mark(X4))) :|: z = 1 + (1 + X4), X4 >= 0
mark(z) -{ 2 }→ a__p(0) :|: z = 1 + 0
mark(z) -{ 2 }→ a__p(1 + mark(X6)) :|: X6 >= 0, z = 1 + (1 + X6)
mark(z) -{ 2 }→ a__p(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
mark(z) -{ 2 }→ a__f(a__p(mark(X''))) :|: z = 1 + (1 + X''), X'' >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(X'))) :|: X' >= 0, z = 1 + (1 + X')
mark(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
mark(z) -{ 2 }→ a__f(1 + mark(X3)) :|: z = 1 + (1 + X3), X3 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(X1') + X2') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 1 }→ 1 + mark(X) :|: z = 1 + X, X >= 0
mark(z) -{ 1 }→ 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2

(11) InliningProof (UPPER BOUND(ID) transformation)

Inlined the following terminating rules on right-hand sides where appropriate:

a__p(z) -{ 1 }→ 0 :|: z = 1 + 0
a__p(z) -{ 1 }→ 1 + X :|: X >= 0, z = X

(12) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
a__f(z) -{ 2 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
a__f(z) -{ 1 }→ 1 + 0 + (1 + (1 + 0)) :|: z = 0
a__p(z) -{ 1 }→ 0 :|: z = 1 + 0
a__p(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
mark(z) -{ 2 }→ a__p(a__p(mark(X5))) :|: X5 >= 0, z = 1 + (1 + X5)
mark(z) -{ 2 }→ a__p(a__f(mark(X4))) :|: z = 1 + (1 + X4), X4 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(X6)) :|: X6 >= 0, z = 1 + (1 + X6)
mark(z) -{ 2 }→ a__p(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
mark(z) -{ 2 }→ a__f(a__p(mark(X''))) :|: z = 1 + (1 + X''), X'' >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(X'))) :|: X' >= 0, z = 1 + (1 + X')
mark(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
mark(z) -{ 2 }→ a__f(1 + mark(X3)) :|: z = 1 + (1 + X3), X3 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(X1') + X2') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 0, X >= 0, 0 = X
mark(z) -{ 1 }→ 1 + mark(X) :|: z = 1 + X, X >= 0
mark(z) -{ 1 }→ 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2

(13) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(14) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
a__f(z) -{ 2 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 1 }→ 1 + 0 + (1 + (1 + 0)) :|: z = 0
a__p(z) -{ 1 }→ 0 :|: z = 1 + 0
a__p(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__p(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
mark(z) -{ 2 }→ a__f(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
mark(z) -{ 2 }→ a__f(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(X1') + X2') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 0, X >= 0, 0 = X
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2

(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

Found the following analysis order by SCC decomposition:

{ a__p } { a__f } { mark }

(16) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
a__f(z) -{ 2 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 1 }→ 1 + 0 + (1 + (1 + 0)) :|: z = 0
a__p(z) -{ 1 }→ 0 :|: z = 1 + 0
a__p(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__p(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
mark(z) -{ 2 }→ a__f(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
mark(z) -{ 2 }→ a__f(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(X1') + X2') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 0, X >= 0, 0 = X
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2

Function symbols to be analyzed: {a__p}, {a__f}, {mark}

(17) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: a__p
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z

(18) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
a__f(z) -{ 2 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 1 }→ 1 + 0 + (1 + (1 + 0)) :|: z = 0
a__p(z) -{ 1 }→ 0 :|: z = 1 + 0
a__p(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__p(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
mark(z) -{ 2 }→ a__f(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
mark(z) -{ 2 }→ a__f(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(X1') + X2') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 0, X >= 0, 0 = X
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2

Function symbols to be analyzed: {a__p}, {a__f}, {mark}
Previous analysis results are:

a__p: runtime: ?, size: O(n1) [1 + z]

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: a__p
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(20) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
a__f(z) -{ 2 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 1 }→ 1 + 0 + (1 + (1 + 0)) :|: z = 0
a__p(z) -{ 1 }→ 0 :|: z = 1 + 0
a__p(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__p(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
mark(z) -{ 2 }→ a__f(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
mark(z) -{ 2 }→ a__f(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(X1') + X2') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 0, X >= 0, 0 = X
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2

Function symbols to be analyzed: {a__f}, {mark}
Previous analysis results are:

a__p: runtime: O(1) [1], size: O(n1) [1 + z]

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(22) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
a__f(z) -{ 2 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 1 }→ 1 + 0 + (1 + (1 + 0)) :|: z = 0
a__p(z) -{ 1 }→ 0 :|: z = 1 + 0
a__p(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__p(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
mark(z) -{ 2 }→ a__f(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
mark(z) -{ 2 }→ a__f(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(X1') + X2') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 0, X >= 0, 0 = X
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2

Function symbols to be analyzed: {a__f}, {mark}
Previous analysis results are:

a__p: runtime: O(1) [1], size: O(n1) [1 + z]

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: a__f
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3 + z

(24) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
a__f(z) -{ 2 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 1 }→ 1 + 0 + (1 + (1 + 0)) :|: z = 0
a__p(z) -{ 1 }→ 0 :|: z = 1 + 0
a__p(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__p(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
mark(z) -{ 2 }→ a__f(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
mark(z) -{ 2 }→ a__f(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(X1') + X2') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 0, X >= 0, 0 = X
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2

Function symbols to be analyzed: {a__f}, {mark}
Previous analysis results are:

a__p: runtime: O(1) [1], size: O(n1) [1 + z] a__f: runtime: ?, size: O(n1) [3 + z]

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: a__f
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 3

(26) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
a__f(z) -{ 2 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 1 }→ 1 + 0 + (1 + (1 + 0)) :|: z = 0
a__p(z) -{ 1 }→ 0 :|: z = 1 + 0
a__p(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__p(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
mark(z) -{ 2 }→ a__f(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(0) :|: z = 1 + 0
mark(z) -{ 2 }→ a__f(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(X1') + X2') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 0, X >= 0, 0 = X
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2

Function symbols to be analyzed: {mark}
Previous analysis results are:

a__p: runtime: O(1) [1], size: O(n1) [1 + z] a__f: runtime: O(1) [3], size: O(n1) [3 + z]

(27) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(28) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 5 }→ s :|: s >= 0, s <= 1 * 0 + 3, z = 1 + 0
a__f(z) -{ 5 }→ s' :|: s' >= 0, s' <= 1 * (1 + (1 + 0)) + 3, z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 1 }→ 1 + 0 + (1 + (1 + 0)) :|: z = 0
a__p(z) -{ 1 }→ 0 :|: z = 1 + 0
a__p(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 5 }→ s'' :|: s'' >= 0, s'' <= 1 * 0 + 3, z = 1 + 0
mark(z) -{ 2 }→ a__p(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
mark(z) -{ 2 }→ a__f(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(X1') + X2') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 0, X >= 0, 0 = X
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2

Function symbols to be analyzed: {mark}
Previous analysis results are:

a__p: runtime: O(1) [1], size: O(n1) [1 + z] a__f: runtime: O(1) [3], size: O(n1) [3 + z]

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: mark
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3·z

(30) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 5 }→ s :|: s >= 0, s <= 1 * 0 + 3, z = 1 + 0
a__f(z) -{ 5 }→ s' :|: s' >= 0, s' <= 1 * (1 + (1 + 0)) + 3, z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 1 }→ 1 + 0 + (1 + (1 + 0)) :|: z = 0
a__p(z) -{ 1 }→ 0 :|: z = 1 + 0
a__p(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 5 }→ s'' :|: s'' >= 0, s'' <= 1 * 0 + 3, z = 1 + 0
mark(z) -{ 2 }→ a__p(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
mark(z) -{ 2 }→ a__f(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(X1') + X2') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 0, X >= 0, 0 = X
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2

Function symbols to be analyzed: {mark}
Previous analysis results are:

a__p: runtime: O(1) [1], size: O(n1) [1 + z] a__f: runtime: O(1) [3], size: O(n1) [3 + z] mark: runtime: ?, size: O(n1) [3·z]

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: mark
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 9 + 36·z

(32) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 5 }→ s :|: s >= 0, s <= 1 * 0 + 3, z = 1 + 0
a__f(z) -{ 5 }→ s' :|: s' >= 0, s' <= 1 * (1 + (1 + 0)) + 3, z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 1 }→ 1 + 0 + (1 + (1 + 0)) :|: z = 0
a__p(z) -{ 1 }→ 0 :|: z = 1 + 0
a__p(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 5 }→ s'' :|: s'' >= 0, s'' <= 1 * 0 + 3, z = 1 + 0
mark(z) -{ 2 }→ a__p(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__p(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
mark(z) -{ 2 }→ a__f(a__p(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(X1') + X2') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 0, X >= 0, 0 = X
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2

Function symbols to be analyzed:
Previous analysis results are:

a__p: runtime: O(1) [1], size: O(n1) [1 + z] a__f: runtime: O(1) [3], size: O(n1) [3 + z] mark: runtime: O(n1) [9 + 36·z], size: O(n1) [3·z]

(33) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(34) BOUNDS(1, n^1)